Subgroups — #10600166

Order 640.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65193	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#64669	C₂	1	2	yes	maximal, complemented
5	D₁₆₀	C₂	1	2	yes	maximal, complemented
6	#4050	—	2	1	no	—
7	C₂×D₄₀	—	2	1	no	—
8	C₂×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₄₀	C₂²	1	1	yes	characteristic, complemented
10	#3932	—	2	1	no	—
11	D₈₀	—	2	1	no	—
12	D₈₀	C₂²	1	2	yes	—
13	D₈₀	C₂²	1	1	yes	characteristic, complemented
14	C₅×C₃₂	C₂²	1	2	yes	complemented
15	#2263	—	5	1	no	maximal, Sylow
16	#709	—	4	1	no	—
17	C₂×D₂₀	—	4	1	no	—
18	C₂×C₅×C₈	D₄	1	1	yes	characteristic
19	Dic₂₀	—	2	1	no	—
20	D₄₀	—	2	2	no	—
21	D₄₀	—	2	1	no	—
22	D₄₀	—	4	1	no	—
23	#677	—	4	1	no	—
24	C₅×C₁₆	D₄	1	1	yes	characteristic
25	C₅×C₁₆	C₂³	1	1	yes	characteristic
26	#508	—	5	1	no	—
27	C₂×D₁₆	—	5	1	no	—
28	SD₆₄	—	5	2	no	—
29	D₃₂	—	5	2	no	—
30	M₆₄	D₅	1	1	yes	characteristic, complemented
31	C₂²×D₅	—	8	1	no	—
32	C₂×C₄×C₅	D₈	1	1	yes	characteristic
33	C₂²⋊D₅	—	8	1	no	—
34	D₂₀	—	4	1	no	—
35	D₂₀	—	4	2	no	—
36	D₂₀	—	8	1	no	—
37	C₄×D₅	—	8	1	no	—
38	Dic₁₀	—	4	1	no	—
39	C₅×C₈	D₈	1	1	yes	characteristic
40	C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	#135	—	10	1	no	—
42	C₂×D₈	—	10	1	no	—
43	Q₃₂	—	5	1	no	—
44	SD₃₂	—	10	1	no	—
45	D₁₆	—	5	2	no	—
46	D₁₆	—	5	1	no	—
47	D₁₆	—	10	1	no	—
48	C₂×C₁₆	C₂×D₅	1	1	yes	characteristic
49	C₃₂	C₂×D₅	1	2	yes	complemented
50	C₂²×C₅	D₁₆	1	1	yes	characteristic
51	C₂×D₅	—	8	1	no	—
52	C₂×D₅	—	8	2	no	—
53	C₂×D₅	—	16	2	no	—
54	C₄×C₅	D₁₆	1	1	yes	characteristic
55	C₄×C₅	C₂×D₈	1	1	yes	characteristic
56	Dic₅	—	8	1	no	—
57	#41	—	20	1	no	—
58	C₂×D₄	—	20	1	no	—
59	Q₁₆	—	10	1	no	—
60	SD₁₆	—	20	1	no	—
61	D₈	—	10	2	no	—
62	D₈	—	10	1	no	—
63	D₈	—	20	1	no	—
64	C₂×C₈	D₂₀	1	1	yes	characteristic
65	C₁₆	D₂₀	1	1	yes	characteristic
66	C₁₆	C₂²×D₅	1	1	yes	characteristic
67	C₂×C₅	—	2	1	no	—
68	C₂×C₅	C₂×D₁₆	1	1	yes	characteristic
69	D₅	—	16	1	no	—
70	D₅	—	16	2	no	—
71	C₂³	—	40	1	no	—
72	Q₈	—	20	1	no	—
73	D₄	—	20	2	no	—
74	D₄	—	20	1	no	—
75	D₄	—	40	1	no	—
76	D₄	—	40	1	no	—
77	C₂×C₄	—	40	1	no	—
78	C₂×C₄	D₄₀	1	1	yes	characteristic
79	C₈	D₄₀	1	1	yes	characteristic
80	C₈	C₂×D₂₀	1	1	yes	characteristic
81	C₅	#2263	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	40	1	no	—
83	C₂²	—	40	2	no	—
84	C₂²	—	80	2	no	—
85	C₂²	D₈₀	1	1	yes	characteristic
86	C₄	—	40	1	no	—
87	C₄	D₈₀	1	1	yes	characteristic
88	C₄	C₂×D₄₀	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	80	2	no	—
91	C₂	—	80	1	no	—
92	C₂	C₂×D₈₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈₀
commutator, cosocle	C₅×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₅
Fitting	C₅×M₆₄	C₂
socle	C₂×C₅	C₂×D₁₆
hypercenter	M₆₄	D₅
hypocenter	C₅	#2263
LCS	G ▷ C₅×C₁₆ ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #65193	C₂	semidirect
2. C₂×D₈₀	C₂	semidirect
3. C₅×M₆₄	C₂	semidirect
4. #64669	C₂	semidirect
5. D₁₆₀	C₂	semidirect
9. Dic₄₀	C₂²	semidirect
13. D₈₀	C₂²	semidirect
14. C₅×C₃₂	C₂²	semidirect
30. M₆₄	D₅	semidirect
49. C₃₂	C₂×D₅	semidirect
81. C₅	#2263	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#65193	2
C₂×D₈₀	2
C₅×M₆₄	2
#64669	2
D₁₆₀	2
#2263	5